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Volume 2009, Article ID 750534, 9 pagesdoi:10.1155/2009/750534 Research Article Localized Mode DFT-S-OFDMA Implementation Using Frequency and Time Domain Interpolation Ari Viholainen,1Te

Volume 2009, Article ID 750534, 9 pages

doi:10.1155/2009/750534

Research Article

Localized Mode DFT-S-OFDMA Implementation Using

Frequency and Time Domain Interpolation

Ari Viholainen,1Tero Ihalainen,1Mika Rinne,2and Markku Renfors (EURASIP Member)1

1 Department of Communications Engineering, Tampere University of Technology, P.O Box 553, 33101 Tampere, Finland

2 Nokia Research Center, P.O Box 407, 00045 Helsinki, Finland

Correspondence should be addressed to Ari Viholainen,ari.viholainen@tut.fi

Received 3 September 2008; Revised 19 December 2008; Accepted 12 March 2009

Recommended by Ana Perez-Neira

This paper presents a novel method to generate a localized mode single-carrier frequency division multiple access (SC-FDMA) waveform Instead of using DFT-spread OFDMA (DFT-S-OFDMA) processing, the new structure called SCiFI-FDMA relies on frequency and time domain interpolation followed by a user-specific frequency shift SCiFI-FDMA can provide signal waveforms that are compatible to DFT-S-OFDMA In addition, it provides any resolution of user bandwidth allocation for the uplink multiple access with comparable computational complexity, because the DFT is avoided Therefore, SCiFI-FDMA allows a flexible choice

of parameters appreciated in broadband mobile communications in the future

Copyright © 2009 Ari Viholainen et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

1 Introduction

OFDMA is a multiple access technique that inherits many

attractive features of the orthogonal frequency division

multiplexing (OFDM) transmission [1, 2] However, as a

multicarrier signal waveform, it suffers from a high

peak-to-average power ratio (PAPR) and hence leads to power

inefficiency, which has serious consequences for an uplink

transmission [3]

Recently single-carrier frequency division

multiple-access (SC-FDMA) transmission by DFT-spread OFDMA

(DFT-S-OFDMA) has drawn increasing attention because

it enables frequency-domain equalization (FDE), advanced

receiver techniques, and low PAPR [4,5] Low PAPR is due

to a serially modulated single-carrier block-transmission,

where the dynamic range of the transmitted signal’s

instan-taneous power is considerably smaller compared to

mul-ticarrier transmission on parallel subcarriers The lower

PAPR reduces the necessary back-off of the nonlinear power

amplifier, required to keep the spectral regrowth and in-band

distortion at a tolerable level This property can be exploited

to improve the cell coverage or to extend the battery

active-time of the mobile terminal

Alternative techniques to generate the SC-FDMA

sig-nal include the DFT-S-OFDMA [6, 7], the DFT-spread

generalized multicarrier (DFT-S-GMC) [8], and the inter-leaved frequency division multiple access (IFDMA) [9] DFT-S-GMC is a frequency-domain technique, where the

M-point IFFT is replaced by an M-band inverse filter bank

transform IFDMA utilizes time-domain generation of the signal waveform using block-wise symbol repetition and a user-specific phase rotation Another interpretation is that IFDMA is equivalent to a distributed mode DFT-S-OFDMA with equidistant subcarrier mapping

DFT-S-OFDMA is a very elegant technique to generate the uplink transmission for a specific user However, the complexity of the DFT and the resulting resolution of the practical DFT sizes are issues for the implementation Thus, a communications standard like the 3GPP E-UTRA [10] limits the supported subset of the DFT sizes K For

bandwidth allocation, the standard defines a resource block

of 12 subcarriers (i.e., K is a multiple of 12 with radix

{2, 3, 5}) The motivation of this paper is to present a generic method to generate SC-FDMA waveform in such a manner that all values ofK are feasible yet with practical complexity.

Here, the IFFT sizeM is assumed to be a power-of-two but

the DFT size K may take all the values from 1 to  M/γ 

with γ being a small positive integer Compared to the

reference design, this method enhances flexibility to allocate bandwidth and provides reduced number of multiplications

for large majority of K values One drawback inherent in

the proposed structure is an approximation error, which

however can nicely be parametrized below the noise level

The rest of this paper is organized as follows InSection 2,

the frequency-domain generation of a single-user signal

based on DFT-S-OFDMA is reviewed Also, the motivation

for developing an alternative implementation solution for

the localized mode is addressed Section 3 describes an

efficient implementation, where initial frequency-domain

interpolation using a specific transform matrix is followed

by fractional time-domain interpolation and user-specific

frequency translation Hereafter, this structure is referred

to as SC-FDMA implementation using frequency and time

domain interpolation (SCiFI-FDMA) The numerical

anal-ysis of DFT-S-OFDMA and SCiFI-FDMA are presented

terms of the number of required multiplications and

addi-tions Section 5 compares the DFT-S-OFDMA and

SCiFI-FDMA techniques to generate an SC-SCiFI-FDMA signal in a

single-user scenario and in a multi-user (multiple access)

scenario The performance is compared by evaluating the

error vector magnitude (EVM) of the approximation error

inherent in the SCiFI-FDMA transmitter due to fractional

interpolation Finally, conclusions are drawn inSection 6

2 Localized Mode DFT-S-OFDMA

DFT-S-OFDMA is a frequency-domain precoding technique

to generate the SC-FDMA signal waveform In this paper, we

focus on the properties of the DFT-IFFT processing shown

and localized mode symbol mapping by simple change of

the subcarrier allocation Both transmission modes provide

signals with an envelope of a single-carrier transmission

This is beneficial in order to minimize the in-band distortion

and out-of-band emissions In general, the localized mode is

preferred in the practical systems due to various

imperfec-tions of the distributed mode The distributed mode signal

has been shown to be sensitive to the carrier frequency offsets

(caused by Doppler effects and/or mismatch of the

transmit-receive oscillators), phase noise, and imperfect power control

[11] The localized mode signal is far less sensitive to these

imperfections

2.1 Signal Model In the signal model of Figure 1, the

discrete Fourier transformed sequence is expressed as

A[k] =

K−1

n =0

x[n]e − j((2π/K)nk), (1)

wherek =0, 1, , K −1 andx[n] is a length-K sequence of

symbols, commonly from a QAM alphabet The subcarrier

allocation specifies how the DFT-spread samples are mapped

to the frequency bins of the IFFT, that is, whether distributed

or localized transmission mode is used The output of the

IFFT is

y[m] = 1

M

M−1

l =0

B[l]e j((2π/M)ml), (2)

. K-point

DFT

.

B

Subcarrier allocation

. M-point

IFFT

.

y

P/S

Figure 1: Frequency-domain realization of SC-FDMA signal

wherem =0, 1, , M −1 andB[l] is the length-M output

sequence of the subcarrier allocation block

In the localized mode, the DFT-spread samples are allocated to a set of contiguous frequency bins of the IFFT This results in the following subcarrier allocation:

B[l] =

⎧

⎨

⎩

A[s], ifl =(s + ξ) mod M,

where s = 0, 1, , K −1, ξ is a user-specific subcarrier

allocation offset (0 ≤ ξ ≤ M −1), and the mod stands for modulo operation The input-output relation of the localized mode DFT-S-OFDMA cannot be simplified as much as in the case of the distributed mode and this results in a higher implementation complexity

2.2 Implementation Complexity In the localized mode, the M-point IFFT can be efficiently implemented via the Split-Radix FFT algorithm requiring M(log2(M) − 3) + 4 real multiplications and 3M(log2(M) −1) + 4 real additions [12] The main challenge is theK-point DFT with arbitrary

values of K The direct computation of the DFT requires

3K2real multiplications This is because one complex mul-tiplication can be calculated with three real mulmul-tiplications and three real additions as shown in [12] A more efficient implementation is possible, if K can be factorized into a

small set of prime numbers Based on this principle, the number of real multiplications required for the K-point

DFT, can be reduced using the Cooley-Tukey algorithm to

3K

i k i l i, where K = i k l i

i [13] Even more efficient techniques, such as Prime Factor and Winograd Fourier transform algorithms, have been reported in the literature [14] However, these highly optimized algorithms are not necessarily practical to the SC-FDMA application due to complicated re-indexing of data and increased memory requirements On the other hand, very low-complexity techniques for a limited set of highly composite DFT sizes up to 1024 have been studied in [15], where the resulting computational complexities have been given in a table format Later on, these specific values are used for comparison and they are referred to as Murphy’s method

multipli-cations when the DFT complexity is estimated using the Cooley-Tukey algorithm and Murphy’s method The differ-ent curves indicate the cases where the DFT size is a prime number, a number with a specific radix representation, and a specific value that is feasible in Murphy’s method,

10 2

10 3

10 4

10 5

10 6

100 200 300 400 500 600 700 800 900 1000

DFT sizeK

Primes

Radix 2,3,5

Radix2,3 Murphy’s method Figure 2: DFT complexity using the Cooley-Tukey algorithm and

Murphy’s method Radix2,3,5means thatK =2l13l25l3, wherel1≥0,

l2≥0, andl3> 0.

respectively As can be seen, the number of required real

multiplications is very high, for example, if the DFT size is a

prime number Murphy’s method provides a low number of

multiplications for a specific set of composite non

power-of-two values ofK with factors 2, 3, or 5 However, the number

of feasibleK values reported in [15] is only 45 while the DFT

size ranges from 10 to 1024

3 SCiFI-FDMA

The motivation for an alternative generic method to generate

a localized mode SC-FDMA signal originates from the

observations discussed inSection 2.2 It is difficult to find an

efficient implementation for arbitrary values of K Our idea

is to apply a novel processing structure, called SCiFI-FDMA,

shown inFigure 3 It is based on the operation by a specific

M1× K transform matrix Q (M1 = γK, where γ ∈ {2, 3}),

a fractional time-domain interpolation, and a user-specific

frequency shift

The matrix multiplication z = Qx can be considered

as the frequency-domain interpolation by theγ-factor This

results in a limitation, where theK value can only be varied

from 1 to  M/γ  Here, the cases of γ = 2 and γ = 3

are studied, which may slightly limit the maximum band

allocation The overcoming of this limitation is considered

as an additional future study

The transform matrix Q is constructed by allocating

the output of the K-point DFT into the first (K + 1)/2 

and the last (K − 1)/2  (here · and · denote ceil

and floor operations, resp.) bins of the M1-point IDFT

This bin allocation results in a very simplified transform

matrix leading to computational savings The time-domain

interpolation increases the number of samples from M1

to the final block length of M samples This calls for a

.

Q

x

z

.

P/S Fractional

interpolation

e jφ[n]

y

Figure 3: An alternative structure (SCiFI-FDMA) for the realiza-tion of the localized mode SC-FDMA signal

fractional interpolation unless M1 is a power-of-two Due

to the proposed bin allocation, the spectrum of a user is centered around the zero-frequency (exactly around zero for odd values ofK and only half a bin to the right of zero for

even values of K) resulting in efficient interpolation The baseband signal format enables fractional interpolation with real-valued anti-imaging filter The last processing element implements the user-specific frequency shift

e jφ[n] = K

Mexp



j2πn M

 K −

1

wheren =0, 1, , M −1 andξ is the subcarrier allocation

offset This frequency shift translates the user spectrum to its scheduled user-specific position in the system band

3.1 Transform Matrix The transform matrix Q defines the

input-output relation of theK-point DFT and the M1-point IDFT for the given zero-centered bin allocation ThisM1× K

matrix can be expressed as

where the elements of theuth row and the vth column in the

DFT and IDFT matrices are defined as

[WDFT]u,v = e − j((2π/K)uv), u, v =0, 1, , K −1,

[WIDFT]u,v = 1

M1e j((2π/M1 )uv), u, v =0, 1, , M1−1,

(6)

respectively Moreover, P denotes anM1× K expansion and

permutation matrix that controls the selection of active bins

The transform matrix Q has an efficient structure when

matrix P is selected according to

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

eT

1

eT

− − − − −

0(M 1− K) × K

− − − − −

eT n+1

eT

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

where 0 is the (M1− K) × K zero matrix, n = (K + 1)/2 ,

and ei =[0· · ·0 1 0· · ·0]T denotes theith K ×1 natural

basis vector (with one as theith element).

With the given IDFT bin allocation, the Q matrix consists

ofγ interlaced K × K circulant matrices Submatrices R i, for

i = 0, 1, , γ −1, can be extracted from the Q matrix by

picking up everyγth row (from top to down) with offsets of

i rows, that is,

[Ri]u,v =[Q]γu+i,v, u, v =0, 1, , K −1. (8)

Furthermore, each of these circulant matrices is fully

charac-terized by its first column vector ri =Rie1 The other column

vectors are obtained as rotations of ri[16] As an example,

the structure of a 3K × K (γ =3) transform matrix is shown

below:

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

a0 a1 · · · a K −2 a K −1

b0 b1 · · · b K −2 b K −1

· · · ·

a K −1 a0 · · · a K −3 a K −2

b K −1 b0 · · · b K −3 b K −2

· · · ·

· · · ·

a1 a2 · · · a K −1 a0

b1 b2 · · · b K −1 b0

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

. (9)

The coefficients on the second row of Q matrix can be

expressed as a v = [R1]0,v forv = 0, 1, , K −1, whereas

the coefficients on the third row of Q matrix are b v =[R2]0,v

The rest of the coefficients are determined by cyclic rotations

according to the definition of a circular matrix

3.1.1 E fficient Implementation of Matrix Multiplication Let

us now consider the transformed vector z It comprises of

γ interlaced subvectors z i(meaning thatz(γu + i) = z i(u)),

each of which is the outcome of the matrix multiplication

zi =Rix can be identified as a circular (cyclic) convolution

between ri and x leads to an efficient frequency-domain

realization The circular convolution is the IDFT of the

product of the DFTs of two length-K vectors [17], that is,

zi =ri x=IDFT(DFT(ri)·DFT(x)). (10)

However, we target to replace the IDFT (DFT) by the IFFT

(FFT) because of the reduced complexity IfK is a

power-of-two then this follows straightforwardly For other values ofK,

vectors riand x are zero-padded (by adding tailing zeros) to

form the length-K vectorsr andx, whereK is the smallest

power-of-two value greater than 2K −1 (Another option

is to use length-K2 DFT, where K2 is the smallest integer value greater than 2K −1, when it can be implemented more efficiently using the Cooley-Tukey algorithm or Murphy’s method) Now,



zi =IFFT(FFT(ri)·FFT(x)) (11) represents actually linear convolution instead of circular convolution Fortunately, there is a relationship between circular convolution and linear convolution as mentioned in [17] Therefore, we can obtain circular convolution by using the following equation:

zi(l) = zi(l) +zi(l + K), l =0, 1, , K −1. (12)

matrix multiplication z = Qx The first branch is simpler

than the others because the first submatrix R0is an identity matrix For other branches, the input vector is first zero-padded to a length of K1 samples and transformed to the frequency domain Then, the resulting vector is multiplied element-wise by a length-K1 vector Φi, which can be obtained by the FFT ofri, and the result is transformed back

to the time domain The L2C-block denotes the conversion from linear convolution to circular convolution Finally, the

vectors ziare interlaced to form the output vector z This is

done by using upsamplers and a delay-chain like in a typical polyphase implementation

It should be pointed out that the coefficients of the vectorsΦi, wherei =1, 2, , γ −1, can be pre-calculated for each desired combination ofK and γ parameters and stored

in a look-up table for run-time access

3.2 Fractional Interpolation Fractional interpolation can

be performed straightforwardly by using a cascade of upsampling by L, image-rejection/anti-alias filtering, and

downsampling by D [18] However, this approach is not the most practical one because it requires a large amount

of unnecessary calculations since only everyDth sample is

finally preserved A more efficient technique for fractional interpolation is to use polynomial-based interpolation filters, that is, filters having a piece-wise polynomial impulse response

The modified Farrow structure, shown in Figure 5, provides an attractive realization for polynomial-based inter-polation filters [19] The input signal is filtered withM p+ 1 parallel real-valued symmetric/antisymmetric FIR filters of order N (with coe fficients c m,n, where m = 0, 1, , M p

and n = 0, 1, , N), to obtain the output samples of

the branch filters C m(z) The interpolated sample value is

obtained by multiplying output samples by so-called basis multipliers (α(μ)) m = (2μ −1)m, for m = 0, 1, , M p, and summing all up Here, M p defines the polynomial-order of the interpolation filter andα(μ) is the

continuous-valued parameter, so-called fractional interval, that controls the time difference between the desired time instant for the interpolated output sample and the previous time instant where the discrete-time input sample exists

γ K

+

M1

z −1

K1

K1 -point FFT

Φ1

× K1 -point

K

.

Φ2

× K1 -point

K γ

z −1

+

.

Φγ−1

× K1 -point

K γ

z −1

Figure 4: Efficient implementation of the matrix multiplication z=Qx is based on the cyclic convolution in the frequency domain.

C M p,0

z −1

z −1

C M

) C M p,N−1

C M p,N

+

+

C2

C1

C0

α(μ)

Figure 5: Modified Farrow structure

3.2.1 Simplified Version of Modified Farrow Structure The

parameters of the modified Farrow structure could be tuned

for different pairs of M1andM in order to obtain the best

tradeoff between performance and complexity However, we

have observed that the polynomial-order of two, M p = 2,

provides sufficiently good performance with arbitrary values

ofM1even if the length of the branch filters is kept relatively

short, that is, N + 1 ≤ 10 Naturally, the branch filter

coefficients have to be pre-optimized and stored in a

look-up table for every pair ofM1andM for run-time execution.

In order to guarantee interpolation where the incoming

values are preserved and new values are generated between

the original ones, the piece-wise polynomial impulse

response should form a Nyquist filter This Nyquist property

results in an additional simplification in the modified Farrow

structure because now different branch filters have a special relationship If themth branch filter is written as

C m(z) =

N



n =0

c m,n z − n, (13)

the filter coefficients are expressed as follows:

c2,n= c2,N− n,

c1,n=

⎧

⎪

⎪

⎪

⎪

0.5, ifn = N −1

2 ,

−0.5, ifn = N + 1

2 ,

0, otherwise,

c0,n=

⎧

⎪

⎪

0.5 − c2,n, ifn = N −1

2 orn = N + 1

2 ,

− c2,n, otherwise.

(14)

Basically, there are only (N + 1)/2 coe fficients, c2,n for

n =0, 1, , (N −1)/2, to be optimized Moreover, it is quite

easy to optimize these coefficients in such a manner that they directly have a sum-of-power-of-two representation This means that a multiplication can be replaced by simple shifts and additions The actual optimization of the filter coefficients is out of the scope of this paper, however, an interested reader can find more details in [20, 21] and references therein

4 Computational Complexity

The computational complexity of DFT-S-OFDMA and SCiFI-FDMA are evaluated by calculating the number of real

multiplications (θ) that is required to compute length-M

complex-valued output sequence The number of real

addi-tions (φ) is also roughly estimated although multiplications

are dominant with regard to the complexity DFT-S-OFDMA

consists of the K-point DFT, subcarrier mapping, and the

M-point IFFT The subcarrier mapping does not require any

multiplications, so the complexity results from the transform

blocks The total number of real multiplications for the

DFT-S-OFDMA structure (assuming the Cooley-Tukey and

the Split-Radix algorithms) is

θ R = θDFT+θIFFT, (15) where

θDFT=

⎛

⎝K

i

k i l i

⎞

⎠ ·3, K =

i

k l i

i,

θIFFT= M

log2(M) −3

+ 4.

(16)

The total number of real multiplications for

SCiFI-FDMA is the sum of the multiplications in each

processing block; the matrix multiplication (θMM), the

modified Farrow structure (θFA), and the user-specific

frequency shift (θFS):

θ S = θMM+θFA+θFS, (17) where

θMM= γ

K1



log2(K1)−3

+ 4

+

γ −1

K1·3,

θFA= M1N + 1

2 ·2 + 2M ·2,

θFS= M ·3.

(18)

The efficient implementation of the matrix multiplication

consists of one FFT and γ −1 IFFTs In addition, γ −1

element-wise length-K1 vector multiplications are required

The modified Farrow structure only consists of (N + 1)/2

different branch filter coefficients (if trivial multiplications

by 0.5 are omitted) and base multipliers α(μ) The frequency

shift requiresM complex multiplications.

In the case of DFT-S-OFDMA, the number of real

additions is calculated using

φ R = φDFT+φIFFT, (19) where

φDFT=

⎛

⎝K

i

k i l i

⎞

⎠ ·3, K =

i

k l i

i,

φIFFT=3M

log2(M) −1

+ 4.

(20)

For SCiFI-FDMA

φ S = φMM+φFA+φFS, (21)

10 4

10 5

10 6

50 100 150 200 250 300 350 400 450 500

K

θ R

θ R2

θ S(γ =2)

θ S(γ =3) Multiple of 12

Figure 6: Number of real multiplications for DFT-S-OFDMA (θR

andθ R2) and SCiFI-FDMA (θS(γ=2) andθ S(γ=3))

where

φMM= γ

3K1



log2(K1)−1

+ 4

+

γ −1

K1·3 +

γ −1

(K −1)·2,

φFA= M1N ·2 +M13·2 +M p M ·2,

φFS= M ·3.

(22)

The following set of parameters is used for the numerical complexity analysis: M = 1024, K = 1, 2, ,  M/γ with

γ ∈ {2, 3},K1is the next power-of-two value greater than

2K −1, M1 = γK, M p = 2, and N = 5 The effect

of the γ-factor on the performance will be discussed in

multiplications and additions as a function of increasingK,

respectively The complexities of DFT-S-OFDMA (θ R,φ R) and SCiFI-FDMA (θ S,φ S) are calculated using (15)–(22) In addition, a number of discrete points (θ R2,φ R2) indicate the DFT-S-OFDMA complexity when the DFT part is estimated using Murphy’s table given in [15] As for theθ Randθ R2, the points that correspond to multiples of 12 are indicated by circle markers

As can be seen, the complexity of DFT-S-OFDMA is a strongly fluctuating function of K when the performance

over the whole range ofK = 1, 2, ,  M/γ is considered Clearly, there are tempting values of K that yield low

complexity, whereas, for example, prime values ofK result

in overwhelming complexity On the other hand, SCiFI-FDMA provides a solution that adds flexibility by allowing moderate and smooth complexity over the whole range of

K values If K is a power-of-two, then the term θMM in (18) is evaluated by substituting K for K1 This results in the downward pointing spikes inFigure 6 The number of

10 4

10 5

10 6

50 100 150 200 250 300 350 400 450 500

K

φ R

φ R2

φ S(γ =2)

φ S(γ =3) Figure 7: Number of real additions for DFT-S-OFDMA (φR and

φ R2) and SCiFI-FDMA (φS(γ=2) andφ S(γ=3))

additions can be quite high for larger values ofK but this is

typically not considered as a problem, because adders are less

costly to implement than multipliers

of DFT-S-OFDMA and SCiFI-FDMA for three sets of K

values The first set (multiples of 12 with radix {2, 3, 5})

clearly favors the DFT-S-OFDMA implementation As for

the second set (other multiples of 12), the relative

per-formance depends on the K value considered and the

difference in complexity fluctuates (+/−) for the benefit

of either structure The third set of arbitrarily chosen

points shows the potential of the SCiFI-FDMA structure

In general, a small set of points that favors either

DFT-S-OFDMA or SCiFI-FDMA can easily be chosen

There-fore, it is necessary to consider the performance over

the full range of K = 1, 2, ,  M/γ  The SCiFI-FDMA

structure is shown to provide lower complexity for 52%

(γ =3) and 81% (γ =2) of cases over the full set ofK values.

When the branch filter coefficients of the modified Farrow

structure have a sum-of-power-of-two representation these

numbers increase to 63% (γ = 3) and 89% (γ = 2),

respectively

Regarding the memory consumption of a practical

implementation, it should be noted that the following

components can be stored in a memory for the run-time

access for each value ofK:

(i)γ −1 vectorsΦiof length-K1,

(ii) (N + 1)/2 pre-optimized Farrow coe fficients c2,n

5 Performance Evaluation

In this section, we compare the DFT-S-OFDMA and

SCiFI-FDMA techniques for implementing an equivalent

Table 1: Number of required multiplications for specific values of

K for DFT-S-OFDMA (θ R/R2) and SCiFI-FDMA (θS(γ=2))

SC-FDMA uplink transmission The comparison is per-formed by evaluating the approximation error introduced by the SCiFI-FDMA transmitter processing The impact of the approximation error is studied both in a single-user case and

in a multi-user case

5.1 Single-User Case We begin the analysis by considering

the approximation error in a single-user case The relevant SCiFI-FDMA parameters for this numerical example are as follows:

N =5, M p =2, γ ∈ {2, 3} (23)

The influence of the approximation error is analyzed through the detection of a SCiFI-FDMA synthesized uplink signal at the receiver side of the link Here, the signal transmission

is assumed to be ideal in a sense that the effects of channel distortion and additive noise are not considered and perfect time synchronization is assumed Moreover, the receiver processing is based on the reference structure (DFT-S-OFDMA) consisting of theM-point FFT, subcarrier

selection, and the K-point IDFT Therefore, the potential

non-idealities of the SCiFI-FDMA processing form the only source of errors in the considered example Figures 8 and

9 show the received signal constellation obtained using the SCiFI-FDMA transmitter with initial frequency-domain upsampling factor of γ = 2 and γ = 3, respectively In the case of DFT-S-OFDMA, the received symbol estimates coincide with the ideal constellation points, whereas for SCiFI-FDMA they disperse slightly around the ideal points due to the approximation error introduced by the time-domain fractional interpolation

The error vector magnitude (EVM) is a well-defined and widely adopted metric to measure the signal quality/purity

−1

−0.5

0

0.5

1

1.5

In-phase

0.3

0.305

0.31

0.315

0.32

0.325

0.33

0.3 0.31 0.32 0.33

In-phase SCiFI-FDMA

16-QAM

Figure 8: Dispersion of the received signal constellation due to

fractional interpolation (γ=2)

In order to estimate the average signal distortion due to the

approximation error, the EVM is evaluated according to [22]:

E

!

!x[n] − x[n]!!2"

E | x[n] |2" ≈

1/N sN s

n =1!!x[n] − x[n]!!2

1/N s

N s

n =1| x[n] |2 ,

(24) where E {·}, x[n], x[n], and N s denote the expectation

of ensemble averages, the actual (measured) and the ideal

symbols, and the length of the symbol sequence, respectively

Moreover, the mean-squared error is normalized by the

average power of the ideal signal It should be emphasized

that the level of EVM can be controlled by adjusting

the SCiFI-FDMA parameters γ, M p, and N This allows

different complexity-performance trade-offs in the actual

system design.Figure 10shows the evaluated (average) EVM

for SCiFI-FDMA with varying DFT size K It can be

observed that the resulting EVM of SCiFI-FDMA is below

−40 dB or−52 dB, over the whole range of K values, for

γ = 2 andγ = 3, respectively Therefore, as the estimated

level of EVM is well below the level of thermal noise

encoun-tered in practise, the BER performance would dominantly be

determined by the SNR operation point instead of the signal

dispersion by the fractional interpolation

5.2 Multi-User Case In a multi-user case, the other users

can be considered as possible sources of multiple access

interference (MAI) due to non-ideal spectral nulls of

SCiFI-FDMA synthesized signals MAI degrades the

detec-tion performance of a specific uplink signal, thus it was

numerically estimated from the received compound signal at

the base station receiver A multiple access reception of ten

simultaneous uplink users with consecutive allocations in the

signal band (with neighboring, non-overlapping frequency

−1.5

−1

−0.5

0

0.5

1

1.5

In-phase

0.3

0.305

0.31

0.315

0.32

0.325

0.33

0.3 0.31 0.32 0.33

In-phase SCiFI-FDMA

16-QAM Figure 9: Dispersion of the received signal constellation due to fractional interpolation (γ=3)

−80

−70

−60

−50

−40

−30

−20

0 50 100 150 200 250 300 350 400 450 500

DFT sizeK

γ =2

γ =3 Figure 10: EVM over a range of the DFT sizeK.

bins) was considered Furthermore, the uplink transmission was assumed to be ideal both in timing and power control From the single-user detection point of view, the other uplink users can be seen as additive noise sources In order

to estimate the variance of MAI, the mean-squared error (MSE) can be estimated, at the frequency bins allocated to

a selected user being detected at a given time, while there are transmissions on the rest of the frequency bins allocated for the rest of the uplink users The average MSE was estimated over a set of one hundred random MA profiles (each with a randomly picked sequence of the DFT sizes and modulation

orders of {4, 16, 64} QAM alphabets for all uplink users).

Moreover, all the considered MA profiles were full bandwidth

scenarios, that is, the DFT sizes allocated for the ten users

summed up to the bandwidth of the IFFT sizeM As a result,

the level of the additive MAI was estimated to be−42 dB and

−52 dB for the design withγ =2 andγ =3, respectively

6 Conclusions

In this paper, SCiFI-FDMA was proposed as a potential

implementation structure for the wideband uplink

trans-mission in a future communication system SCiFI-FDMA

is based on frequency and time domain interpolation

and a user-specific frequency shift It was shown that the

SCiFI-FDMA structure is able to generate signal waveforms

comparable to those obtained with DFT-S-OFDMA The

main advantages of SCiFI-FDMA are its enhanced flexibility

to the generic choice of allocated bandwidth per user and its

competitive computational complexity

In this paper, the performance was analyzed using

exper-imentally chosen parameter values to satisfy the expected

requirements of a communication system Naturally, the

parameters can further be fine-tuned and filters re-optimized

depending on the targeted performance Based on its

charac-teristics, the SCiFI-FDMA offers attractive trade-offs for the

synthesis of SC-FDMA waveforms

Acknowledgments

This research was supported by Nokia (project Waveform

Analysis for Cellular Systems) Moreover, Tero Ihalainen

would like to thank Tampere Graduate School in

Informa-tion Science and Engineering (TISE) for financial support

during this research

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